Improved Bound for the Gerver-Ramsey Collinearity Problem

Abstract

Let $S$ be a finite subset of ${\mathbb{Z}}^n$. A vector sequence $({\mathbf{z}}_i)$ is an $S$-walk if and only if ${\mathbf{z}}_{i+1}-{\mathbf{z}}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\subset {\mathbb{Z}}^3$ there exists an infinite $S$-walk in which no $5^{11}+1=48,828,126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.

22 pages, 4 figures